Lemma 6.12 (See [1]}.)
 Referring to Definition REF ,
there exists a sequence \(\lbrace \varphi _i\rbrace _{i=1}^d\)  of scalars in \(\mathbb {F}\) 
and a basis of \(V\)  with respect to which the matrices representing \(A\)  and \(A^*\)  are
\(A &:\begin{pmatrix}\theta _0 & & & & & \text{\bf 0} \\1 & \theta _1 \\& 1 & \theta _2 \\& & \cdot & \cdot \\& & & \cdot & \cdot \\\text{\bf 0} & & & & 1 & \theta _d\end{pmatrix},&A^* &:\begin{pmatrix}\theta ^*_0 & \varphi _1 & & & & {\bf 0} \\& \theta ^*_1 & \varphi _2 \\& & \theta ^*_1 & \cdot \\& & & \cdot & \cdot \\& & & & \cdot & \varphi _d \\\text{\bf 0} & & & & & \theta ^*_d\end{pmatrix}.\) 
